Ceres主要是为了解决大规模BA(Bundle Adjustment)问题的。
BA是通过最小化重投影误差来同时优化3D点和相机位姿的。
1. BA问题的模型
假设世界坐标系到相机坐标系的变换为： 旋转 R，平移 t
相机的参数为：焦距 f ，畸变参数 k1、k2
所以3D点X转换为图像坐标的过程如下：
- P = R * X + t （从世界坐标系转化到相机坐标系）
- p = -P / P.z （归一到z=1平面，为下面转化到图像坐标系做准备）
- p' = f * r(p) * p （转换到图像坐标系下，考虑了畸变）
其中 r(p) = 1.0 + k1 * ||p||^2 + k2 * ||p||^4
所以考虑相机的畸变，3D点X到图像坐标p'是个非线性变换。
2. 根据上述模型定义代价函数
定义代价函数如下：
struct SnavelyReprojectionError {
SnavelyReprojectionError(double observed_x, double observed_y)
: observed_x(observed_x), observed_y(observed_y) {}
template
bool operator()(const T* const camera,
const T* const point,
T* residuals) const {
// camera[0,1,2] are the angle-axis rotation.
T p[3];
ceres::AngleAxisRotatePoint(camera, point, p);
// camera[3,4,5] are the translation.
p[0] += camera[3];
p[1] += camera[4];
p[2] += camera[5];
// Compute the center of distortion. The sign change comes from
// the camera model that Noah Snavely's Bundler assumes, whereby
// the camera coordinate system has a negative z axis.
T xp = - p[0] / p[2];
T yp = - p[1] / p[2];
// Apply second and fourth order radial distortion.
const T& l1 = camera[7];
const T& l2 = camera[8];
T r2 = xp*xp + yp*yp;
T distortion = 1.0 + r2 * (l1 + l2 * r2);
// Compute final projected point position.
const T& focal = camera[6];
T predicted_x = focal * distortion * xp;
T predicted_y = focal * distortion * yp;
// The error is the difference between the predicted and observed position.
residuals[0] = predicted_x - observed_x;
residuals[1] = predicted_y - observed_y;
return true;
}
// Factory to hide the construction of the CostFunction object from
// the client code.
static ceres::CostFunction* Create(const double observed_x,
const double observed_y) {
return (new ceres::AutoDiffCostFunction(
new SnavelyReprojectionError(observed_x, observed_y)));
}
double observed_x;
double observed_y;
};
其中observed_x和observed_y为与3D点对应的已知的图像图标，为观测值，predicted_x和predicted_x是根据上述模型求出的图像坐标，为计算值。残差是重投影误差，即是他们之间的差。
通过最小化重投影误差，同时优化3D点和相机位姿。
这里定义了一个静态函数Create来方便实例化SnavelyReprojectionError对象，使用的是自动求导的方式。
3. Ceres 求解过程
完整的 Ceres 求解过程如下：
3.1 定义代价函数
上面定义了代价函数SnavelyReprojectionError。
3.2 构建问题，添加残差项
ceres::Problem problem;
for (int i = 0; i < bal_problem.num_observations(); ++i) {
ceres::CostFunction* cost_function =
SnavelyReprojectionError::Create(observations[2 * i + 0],
observations[2 * i + 1]);
problem.AddResidualBlock(cost_function,
NULL /* squared loss */,
bal_problem.mutable_camera_for_observation(i),
bal_problem.mutable_point_for_observation(i));
}
bal_problem.mutable_camera_for_observation(i)和bal_problem.mutable_point_for_observation(i)分别是从文件里读取的相机位姿和3D点，observations数组里则是从文件里去读的与3D点对应的图像坐标点。
3.3 定义求解器的选项参数，并求解
ceres::Solver::Options options;
options.linear_solver_type = ceres::DENSE_SCHUR;
options.minimizer_progress_to_stdout = true;
ceres::Solver::Summary summary;
ceres::Solve(options, &problem, &summary);
std::cout << summary.FullReport() << "\n";
求解器类型用了ceres::DENSE_SCHUR,因为对于BA问题有特殊的稀疏性结构的特点，ceres::DENSE_SCHUR比ceres::DENSE_QR和SPARSE_NORMAL_CHOLESKY都合适
4. 完整代码
#include
#include
#include
#include "ceres/ceres.h"
#include "ceres/rotation.h"
// Read a Bundle Adjustment in the Large dataset.
class BALProblem {
public:
~BALProblem() {
delete[] point_index_;
delete[] camera_index_;
delete[] observations_;
delete[] parameters_;
}
int num_observations() const { return num_observations_; }
const double* observations() const { return observations_; }
double* mutable_cameras() { return parameters_; }
double* mutable_points() { return parameters_ + 9 * num_cameras_; }
double* mutable_camera_for_observation(int i) {
return mutable_cameras() + camera_index_[i] * 9;
}
double* mutable_point_for_observation(int i) {
return mutable_points() + point_index_[i] * 3;
}
bool LoadFile(const char* filename) {
FILE* fptr = fopen(filename, "r");
if (fptr == NULL) {
return false;
};
FscanfOrDie(fptr, "%d", &num_cameras_);
FscanfOrDie(fptr, "%d", &num_points_);
FscanfOrDie(fptr, "%d", &num_observations_);
point_index_ = new int[num_observations_];
camera_index_ = new int[num_observations_];
observations_ = new double[2 * num_observations_];
num_parameters_ = 9 * num_cameras_ + 3 * num_points_;
parameters_ = new double[num_parameters_];
for (int i = 0; i < num_observations_; ++i) {
FscanfOrDie(fptr, "%d", camera_index_ + i);
FscanfOrDie(fptr, "%d", point_index_ + i);
for (int j = 0; j < 2; ++j) {
FscanfOrDie(fptr, "%lf", observations_ + 2*i + j);
}
}
for (int i = 0; i < num_parameters_; ++i) {
FscanfOrDie(fptr, "%lf", parameters_ + i);
}
return true;
}
private:
template
void FscanfOrDie(FILE *fptr, const char *format, T *value) {
int num_scanned = fscanf(fptr, format, value);
if (num_scanned != 1) {
LOG(FATAL) << "Invalid UW data file.";
}
}
int num_cameras_;
int num_points_;
int num_observations_;
int num_parameters_;
int* point_index_;
int* camera_index_;
double* observations_;
double* parameters_;
};
// Templated pinhole camera model for used with Ceres. The camera is
// parameterized using 9 parameters: 3 for rotation, 3 for translation, 1 for
// focal length and 2 for radial distortion. The principal point is not modeled
// (i.e. it is assumed be located at the image center).
struct SnavelyReprojectionError {
SnavelyReprojectionError(double observed_x, double observed_y)
: observed_x(observed_x), observed_y(observed_y) {}
template
bool operator()(const T* const camera,
const T* const point,
T* residuals) const {
// camera[0,1,2] are the angle-axis rotation.
T p[3];
ceres::AngleAxisRotatePoint(camera, point, p);
// camera[3,4,5] are the translation.
p[0] += camera[3];
p[1] += camera[4];
p[2] += camera[5];
// Compute the center of distortion. The sign change comes from
// the camera model that Noah Snavely's Bundler assumes, whereby
// the camera coordinate system has a negative z axis.
T xp = - p[0] / p[2];
T yp = - p[1] / p[2];
// Apply second and fourth order radial distortion.
const T& l1 = camera[7];
const T& l2 = camera[8];
T r2 = xp*xp + yp*yp;
T distortion = 1.0 + r2 * (l1 + l2 * r2);
// Compute final projected point position.
const T& focal = camera[6];
T predicted_x = focal * distortion * xp;
T predicted_y = focal * distortion * yp;
// The error is the difference between the predicted and observed position.
residuals[0] = predicted_x - observed_x;
residuals[1] = predicted_y - observed_y;
return true;
}
// Factory to hide the construction of the CostFunction object from
// the client code.
static ceres::CostFunction* Create(const double observed_x,
const double observed_y) {
return (new ceres::AutoDiffCostFunction(
new SnavelyReprojectionError(observed_x, observed_y)));
}
double observed_x;
double observed_y;
};
int main(int argc, char** argv) {
google::InitGoogleLogging(argv[0]);
if (argc != 2) {
std::cerr << "usage: simple_bundle_adjuster \n";
return 1;
}
BALProblem bal_problem;
if (!bal_problem.LoadFile(argv[1])) {
std::cerr << "ERROR: unable to open file " << argv[1] << "\n";
return 1;
}
const double* observations = bal_problem.observations();
// Create residuals for each observation in the bundle adjustment problem. The
// parameters for cameras and points are added automatically.
ceres::Problem problem;
for (int i = 0; i < bal_problem.num_observations(); ++i) {
// Each Residual block takes a point and a camera as input and outputs a 2
// dimensional residual. Internally, the cost function stores the observed
// image location and compares the reprojection against the observation.
ceres::CostFunction* cost_function =
SnavelyReprojectionError::Create(observations[2 * i + 0],
observations[2 * i + 1]);
problem.AddResidualBlock(cost_function,
NULL /* squared loss */,
bal_problem.mutable_camera_for_observation(i),
bal_problem.mutable_point_for_observation(i));
}
// Make Ceres automatically detect the bundle structure. Note that the
// standard solver, SPARSE_NORMAL_CHOLESKY, also works fine but it is slower
// for standard bundle adjustment problems.
ceres::Solver::Options options;
options.linear_solver_type = ceres::DENSE_SCHUR;
options.minimizer_progress_to_stdout = true;
ceres::Solver::Summary summary;
ceres::Solve(options, &problem, &summary);
std::cout << summary.FullReport() << "\n";
return 0;
}
5. 总结
总结使用步骤如下：
- 定义代价函数。根据BA问题模型定义代价函数
- 构建问题。添加残差项。这里要根据传感器获取的3D点、图像点、和粗略计算的相机位姿，不断地添加残差项。
- 求解问题。定义求解器的选项参数和求解器的结果报告
6. 参考
http://ceres-solver.org/nnls_tutorial.html#bundle-adjustment
http://grail.cs.washington.edu/projects/bal/